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National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
4
Standard NT
Standard NT
Source: 2024 CTST P4
March 7, 2024
number theory
2024 CTST
China TST
Problem Statement
Let
n
n
n
be a positive square free integer,
S
S
S
is a subset of
[
n
]
:
=
{
1
,
2
,
…
,
n
}
[n]:=\{1,2,\ldots ,n\}
[
n
]
:=
{
1
,
2
,
…
,
n
}
such that
∣
S
∣
≥
n
/
2.
|S|\ge n/2.
∣
S
∣
≥
n
/2.
Prove that there exists three elements
a
,
b
,
c
∈
S
a,b,c\in S
a
,
b
,
c
∈
S
(can be same), satisfy
a
b
≡
c
(
m
o
d
n
)
.
ab\equiv c\pmod n.
ab
≡
c
(
mod
n
)
.
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